# Quantum Teleportation Homework: Deriving EPR Pair & Measuring Spin 1/2 Particles

• Markus Kahn
In summary, quantum teleportation is a process where the exact state of a particle is transferred from one location to another through the principles of quantum entanglement. This is achieved by entangling two particles and replicating the state of one particle onto the other through measurement. An EPR pair is a pair of entangled particles, first proposed by Einstein, Podolsky, and Rosen. To measure spin 1/2 particles, a device called a Stern-Gerlach apparatus can be used to separate particles with different spin values.
Markus Kahn

## Homework Statement

This isn't exactly a problem but rather a problem in understanding the derivation of the phenomenon, or more precisely, one step in the derivation.

In the following we will consider the EPR pair of two spin ##1/2## particles, where the state can be written as
$$\vert \psi\rangle =\frac{1}{\sqrt{2}}(\vert 0,1\rangle - \vert 1,0\rangle).$$ Now let us assume that Alice and Bob have each one of the two particles of the EPR pair. Alice has another particle with spin ##1/2## in the state ##\vert \phi\rangle##. The state of the whole system, all three particles, is therefore given by
\begin{align*}\vert \phi\rangle \otimes \vert \psi\rangle &= \vert \phi\rangle \otimes \frac{1}{\sqrt{2}}(\vert 0,1\rangle - \vert 1,0\rangle)\\ &= \frac{1}{\sqrt{2}} (\vert\phi,0\rangle \otimes \vert 1\rangle - \vert\phi,1\rangle \otimes \vert 0\rangle). \end{align*} Now Alice can measure her two particles, for example using ##P_i= \vert \chi_i\rangle\langle \chi_i\vert, i\in \{1,2,3,4\}## and
\begin{align*} \vert\chi_1\rangle &= \frac{1}{\sqrt{2}}(\vert 0,1\rangle - \vert 1,0\rangle)\\ \vert\chi_2\rangle &= \frac{1}{\sqrt{2}}(\vert 0,1\rangle + \vert 1,0\rangle)\\ \vert\chi_1\rangle &= \frac{1}{\sqrt{2}}(\vert 0,0\rangle - \vert 1,1\rangle)\\ \vert\chi_1\rangle &= \frac{1}{\sqrt{2}}(\vert 0,0\rangle + \vert 1,1\rangle). \end{align*}
Up until this point I understand the definitions and the idea. The problem arises when I try to calculate for example
$$P_1 \vert \phi\rangle\otimes\vert\psi\rangle = \frac{1}{2} \vert \chi_1\rangle \otimes (-\vert 1\rangle\langle 1\vert\phi\rangle - \vert 0\rangle\langle 0\vert\phi\rangle )$$

All given above.

## The Attempt at a Solution

We first need to figure out how ##P_i## acts on the tensor product of the states. Expanding the state gives
$$P_1 \vert \phi\rangle\otimes\vert\psi\rangle = \frac{1}{\sqrt{2}} P_1(\vert\phi\rangle \otimes\vert 0\rangle \otimes \vert 1\rangle - \vert\phi\rangle \otimes\vert 1\rangle \otimes \vert 0\rangle).$$ Form this we can conclude that ##P_i## is of the form ##P_i = A\otimes B \otimes C##, where ##A,B,C## can be any operator. I tried to compute now the follwoing:
\begin{align*}\vert \chi_1\rangle\langle \chi_1\vert &= \frac{1}{2}(\vert 0,1\rangle - \vert 1,0\rangle)(\langle 0,1 \vert -\langle 1,0\vert)\\ &= \frac{1}{2} (\vert 0\rangle \otimes\vert1\rangle - \vert 1\rangle \otimes\vert0\rangle)(\langle 0\vert\otimes\langle1 \vert -\langle 1\vert\otimes\langle0\vert), \end{align*}
but can't really proceed from here since I don't really know how to calculate this... I suspect that after finishing this calculation I could define ##A\otimes B := \vert \chi_1\rangle\langle \chi_1\vert ##. Then I would only need to find ##C##, but I'm not really sure how to do that...

Am I doing something completely wrong here, or is this the right approach?

Using that Alice has particles 1 and 2 and Bob particle 3, and particles 2 and 3 are the EPR pair, then the ##P_i## are projection operators for particles 1 and 2 only, with an identity operation for particle 3,
$$(P_i)_{12} \otimes I_3$$
For example, if the three particles are in the state
$$| \Psi \rangle = \frac{1}{\sqrt{2}} \left( |0 \rangle_1 |1 \rangle_2 |0 \rangle_3 + |1 \rangle_1 |0 \rangle_2 |1 \rangle_3 \right)$$
then
$$P_1 | \Psi \rangle = \frac{1}{\sqrt{2}} | \chi_1 \rangle_{12} \left( |0 \rangle_3 - |1 \rangle_3 \right)$$

Thank you very much for the suggestion, but I'm still not sure if I'm doing the math right... Could you maybe just glance over this and tell me if the individual steps work?
\begin{align*}P_1\vert\phi\rangle\otimes\vert \psi\rangle &= \left(\vert \chi_1\rangle\langle\chi_1\vert\otimes I\right) \left( \vert\phi\rangle\otimes\vert\psi\rangle\right) \\ &= \left(\vert \chi_1\rangle\langle\chi_1\vert\otimes I\right) \frac{1}{\sqrt{2}} (\vert\phi,0\rangle \otimes \vert 1\rangle - \vert\phi,1\rangle \otimes \vert 0\rangle) \\ &= \frac{1}{\sqrt{2}} \left(\vert \chi_1\rangle\langle\chi_1\vert \phi,0\rangle \otimes \vert 1\rangle- \vert \chi_1\rangle\langle\chi_1\vert\phi,1\rangle \otimes \vert 0\rangle\right)\\ &= \frac{1}{2} \left([ \underbrace{\langle 0,1\vert \phi,0\rangle}_{=0} - \langle 1,0\vert \phi,0\rangle] \vert \chi_1\rangle \otimes \vert 1\rangle- [ \langle 0,1\vert \phi,1\rangle - \underbrace{\langle 1,0\vert \phi,1\rangle}_{=0}]\vert \chi_1\rangle \otimes \vert 0\rangle\right)\\ &= \frac{1}{2}\left(-\langle 1\vert\phi\rangle \vert\chi_1\rangle\otimes \vert 1\rangle -\langle 0\vert\phi\rangle \vert\chi_1\rangle\otimes \vert 0\rangle\right)\\ &= \frac{1}{2}\vert\chi_1\rangle \otimes \left( -\langle 1\vert\phi\rangle \vert 1\rangle -\langle 0\vert\phi\rangle \vert 0\rangle \right) \\ &= \frac{1}{2}\vert\chi_1\rangle \otimes \left( -\vert 1\rangle\langle 1\vert\phi\rangle -\vert 0\rangle\langle 0\vert\phi\rangle \right) \end{align*}

Markus Kahn said:
\begin{align*}P_1\vert\phi\rangle\otimes\vert \psi\rangle &= \left(\vert \chi_1\rangle\langle\chi_1\vert\otimes I\right) \left( \vert\phi\rangle\otimes\vert\psi\rangle\right) \\ &= \left(\vert \chi_1\rangle\langle\chi_1\vert\otimes I\right) \frac{1}{\sqrt{2}} (\vert\phi,0\rangle \otimes \vert 1\rangle - \vert\phi,1\rangle \otimes \vert 0\rangle) \end{align*}
This is very hard to follow. It is not clear which are single-particle kets and which are two-particle kets. Otherwise, it looks correct.

## 1. What is quantum teleportation?

Quantum teleportation is a process in which the exact state of a particle (such as its spin or position) is transferred from one location to another, without physically moving the particle itself. This is made possible through the principles of quantum entanglement.

## 2. How does quantum teleportation work?

In quantum teleportation, two particles are entangled, meaning that they share a correlated state. The state of one particle, known as the "qubit", is then transferred to the other particle through a process called "measurement". This allows for the state of the first particle to be replicated in the second particle, essentially "teleporting" the information.

## 3. What is an EPR pair?

An EPR (Einstein-Podolsky-Rosen) pair is a pair of particles that are entangled, meaning that they share a correlated state. This was first proposed by Albert Einstein, Boris Podolsky, and Nathan Rosen in 1935 as an example of quantum entanglement.

## 4. How do you derive an EPR pair?

An EPR pair can be derived through a process of entanglement, where two particles are placed in a specific state that allows for their states to be correlated. This can be done through various quantum operations, such as the controlled-NOT (CNOT) gate.

## 5. How do you measure spin 1/2 particles?

The spin of a particle is a quantum property that describes its intrinsic angular momentum. To measure the spin of a particle, you can use a device called a Stern-Gerlach apparatus, which uses a magnetic field to separate particles with different spin values. This allows for the measurement of the spin of a particle to be determined.

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